Remark 3 Using the Envelope theorem (see for instance [1]}) we see that \(H_1\)  and \(H_2\)  being \(\mathcal {C}^1\)  —Assumption REF — in the \(p,M\) -variables implies that, for any couple \((a^1(x,p), a^2(x,M))\)  such that \(H_1(x,p) = b(x,a^1(x,p)).p - f_1(x,a^1(x,p))\)  and \(H_2(x,M) = \sigma ^T\sigma (x, a^2(x,M)).M - f_2(x,a^2(x,M))\)  we get \(\partial _pH_1(x,p) = b(x,a^1(x,p))\)  and \(\partial _MH_2(x,M) = \sigma ^T\sigma (x, a^2(x,M))\) . Consequently, drift and volatility must agree on potentially different optimal controls with common values \(\partial _pH_1(x,p)\)  and \(\sqrt{\partial _MH_2(x,M)}\)  respectively. Notice that the growth conditions on the cost \(f_1\)  and the drift \(b\) , together with the compactness of \(A_2\)  ensure that for any \((x,p,M) \in \mathbb {R}^d \times \mathbb {R}^d \times \mathbb {S}^d(\mathbb {R}) \) , there exists at least one such couple \((a^1(x,p), a^2(x,M)) \in A_1 \times A_2\) .

